Disconjugacy of solutions of a~second order differential equation with Colombeau generalized functions in coefficients

I. G. Kim

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 1, pp. 21-28

Résumé

We consider a differential equation \begin{equation} Lx\doteq x''+P(t)x'+Q(t)x=0,\qquad t\in[a, b]\subset\mathcal I\doteq(\alpha,\beta)\subset\mathbb R, \end{equation} where $P,Q$ are $C$-generalized functions defined on $\mathcal I$ and are known as equivalence classes of Colombeau algebra. Let $\mathcal R_P$ and $\mathcal R_Q$ be representatives of $P$ and $Q$ respectively, $\mathcal A_N$ are classes of functions with compact support used to define Colombeau algebra. We obtain new sufficient conditions for disconjugacy of the equation (1). We prove that if the condition \begin{equation*} (\exists N\in\mathbb N)\,(\forall\varphi\in\mathcal A_N)\,(\exists\mu_01)\ \int_a^b|\mathcal R_P(\varphi_\mu,t)|\,dt+\int_a^b|\mathcal R_Q(\varphi_\mu,t)|\,dt\frac4{b-a+4}\quad(0\mu\mu_0), \end{equation*} is satisfied, where $\varphi_\mu\doteq\frac1\mu\varphi\left(\frac t\mu\right)$, then the equation (1) is disconjugate on $[a,b]$. We prove the separation theorem and its corollary.

Détail

Keywords: $C$-generalized function, $C$-generalized number, weak equation, disconjugacy.